Properties

Label 405600.ca
Number of curves $4$
Conductor $405600$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("ca1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 405600.ca

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.ca1 405600ca4 \([0, -1, 0, -4399633, -3550512863]\) \(30488290624/195\) \(60238576320000000\) \([2]\) \(6193152\) \(2.4044\)  
405600.ca2 405600ca2 \([0, -1, 0, -914008, 273851512]\) \(2186875592/428415\) \(16543019021880000000\) \([2]\) \(6193152\) \(2.4044\) \(\Gamma_0(N)\)-optimal*
405600.ca3 405600ca1 \([0, -1, 0, -280258, -53163488]\) \(504358336/38025\) \(183539412225000000\) \([2, 2]\) \(3096576\) \(2.0578\) \(\Gamma_0(N)\)-optimal*
405600.ca4 405600ca3 \([0, -1, 0, 268992, -236612988]\) \(55742968/658125\) \(-25413149385000000000\) \([2]\) \(6193152\) \(2.4044\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 405600.ca1.

Rank

sage: E.rank()
 

The elliptic curves in class 405600.ca have rank \(0\).

Complex multiplication

The elliptic curves in class 405600.ca do not have complex multiplication.

Modular form 405600.2.a.ca

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{9} - 2 q^{17} - 4 q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.